Dynamics on the space of 2-lattices in 3-space

Abstract: We study the dynamics of $SL_3(\mathbb{R})$ and its subgroups on the homogeneous space $X$ consisting of homothety classes of rank-2 discrete subgroups of $\mathbb{R}^3$. We focus on the case where the acting group is Zariski dense in either $SL_3(\mathbb{R})$ or $SO(2,1)(\mathbb{R})$. Using techniques of Benoist and Quint we prove that for a compactly supported probability measure $\mu$ on $SL_3(\mathbb{R})$ whose support generates a group which is Zariski dense in $SL_3(\mathbb{R})$, there exists a unique $\mu$-stationary probability measure on $X$. When the Zariski closure is $SO(2,1)(\mathbb{R})$ we establish a certain dichotomy regarding stationary measures and discover a surprising phenomenon: The Poisson boundary can be embedded in $X$. The embedding is of algebraic nature and raises many natural open problems. Furthermore, motivating applications to questions in the geometry of numbers are discussed.